## Research Interests

Research Interests

My main research interests are in nonlinear elliptic and parabolic partial differential equations and applications in geometry and optimal transportation. In particular, the regularity theory of Monge-Ampère equations, Hessian equations, and other variational problems.

I am also interested in related areas of geometry and physics, including geometry of convex bodies, minimal surfaces, surfaces of prescribed curvatures, and geometric flows.

### My research profile can be accessed via the following links:

My research profile can be accessed via the following links:

## Preprints

Preprints

, with Y. Huang, submitted. Available at arXiv:1812.03309.**Noncompact $L_p$-Minkowski problems**, with S. Chen; Y. Huang and Q.-R. Li. Available at arXiv:1811.10181.**$L_p$-Brunn-Minkowski inequality for $p\in(p_0,1)$**, with G. Loeper, submitted. Available at arXiv:1809.07432.**Optimal transport with discrete long range mean field interactions**with S. Chen and X.-J. Wang, submitted. Available at arXiv:1806.09482.**Boundary regularity for the second boundary-value problem of Monge-Ampère equations in dimension two,**, with S. Chen and X.-J. Wang, submitted. Available at arXiv:1802.07518.**Global regularity for the Monge-Ampère equation with natural boundary condition**

## Publications (in mathematics)

Publications (in mathematics)

, with S. Chen and X.-J. Wang.**Global regularity of optimal mappings in non-convex domains***SCIENCE CHINA Mathematics*, accepted., with K. Du and F. Zhang.**Stochastic continuity of random fields governed by a system of stochastic PDEs***Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques,*accepted. Available at arXiv:1706.01588., with T.-V. Khanh and P.-T. Thuc.**Bergman-Toeplitz operators on fat Hartogs triangles***Proceedings of the American Mathematical Society,*147 (2019), 327--338. Available at arXiv:1802.09174., with T.-V. Khanh and P.-T. Thuc.**Bergman-Toeplitz operators on weakly pseudoconvex domains***Mathematische Zeitschrift*, 291 (2019), 591--607. Available at arXiv:1710.10761., with K. Du.**On the Cauchy problem for stochastic parabolic equations in Hölder spaces***Transaction of the American Mathematical Society*, 371 (2019), 2643--2664. Available at arXiv:1511.02573., with G. Loeper.**Optimal transport with discrete mean field interaction***2017 MATRIX Annals, Editors: David R. Wood, Jan de Gier, Cheryl E. Praeger, Terence Tao.*MATRIX Book Series, Volume 2,*Springer*, 2018.. Surveys in Geometric Analysis 2017,**Schauder estimates and application on stochastic partial differential equations***Editors: Gang Tian, Qing Han, Zhenlei Zhang,*Science Press, pp.94--115, 2018., with Y.Y. Li and L. Nguyen.**A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions***Journal of Fixed Point Theory and Application*, 19 (2017), 853--876., with K. Du.**A Schauder estimate for stochastic PDEs***C. R. Math. Acd. Sci. Paris, Ser. I*, 354 (2016), 371--375., with N.S. Trudinger.**On the classical solvability of near field reflector problems***Discrete Contin. Dyn. Syst.*, 36 (2016), 895--916.with Y. Huang and L. Xu.**On the uniqueness of $L_p$-Minkowski problems: The constant $p$-curvature case in $\mathbb{R}^3$,***Advances in Mathematics*, 281 (2015), 906--927. Available at arXiv:1503.02358., with Y. Huang and F. Jiang.**Boundary $C^{2,\alpha}$ estimates for Monge-Ampère type equations***Advances in Mathematics*, 281 (2015), 706--733., with N.S. Trudinger and X.-J. Wang.**On asymptotic behaviour and $W^{2,p}$ regularity of potentials in optimal transportation***Arch. Rat. Mech. Anal.*, 215 (2015), 867--905., with X.-J. Wang.**Interior a priori estimates for the Monge-Ampère equation***Surveys in Differential Geometry*Volume 19,*International Press*(2014), 151--177., with S.-Y. A. Chang and P. Yang.**Optimal transportation on the hemisphere***Bulletin of the Institute of Mathematics, Academia Sinica*9 (2014), 25--44.(with Y. Y. Li and L. Nguyen).**Degree theory for oblique boundary problems,***Oberwolfach Report*40/2013:26--29., with B. Zhou.**An obstacle problem for a class of fourth order equations***Journal of Differential Equations*254 (2013), 1306--1325..**Light reflection is nonlinear optimization***Calc. Var. Partial Differential Equations*46 (2013), 861--878.**Regularity of Monge-Ampère equations in optimal transportation.***Bull. Austral. Math. Soc.*83 (2011), 173--176., with N.S. Trudinger.**On Pogorelov estimates for Monge-Ampère type equations***Discrete Contin. Dyn. Syst.*28 (2010), no. 3, 1121--1135., with N.S. Trudinger and X.-J. Wang.**Interior $C^{2,\alpha}$ regularity for potential functions in optimal transportation***Comm. in Partial Differential Equations*35 (2010), 165--184.(with N. S. Trudinger and X.-J. Wang).**Regularity in optimal transportation,***Oberwolfach Report*36/2009:17--20.**Hölder regularity of optimal mappings in optimal transportation.***Calc. Var. Partial Differential Equations*34 (2009), no. 4, 435--451.

## Publications (in computer science)

Publications (in computer science)

, with J. Zhang; M. Zhang and F. Ren.**A multiagent-based domain transportation approach for optimal resource allocation in emergency management***Multi-agent and Complex Systems*, pages 19--32, Springer, Singapore, 2017., with J. Zhang; M. Zhang and F. Ren.**A decentralised multi-agent system for emergency resource allocation in metropolitan regions***Proceedings of the 2016 International Conference on Autonomous Agents & Multiagent Systems*, pages 1482--1484, 2016.with J. Zhang; M. Zhang and F. Ren.**Enable efficient resource deployment in multiple concurrent emergency events through a decentralised MAS,***Australasian Joint Conference on Artificial Intelligence*, pages 56--68, 2016.